Steps in Differential Geometry, Proceedings of the Colloquium
on Differential Geometry, 25–30 July, 2000, Debrecen, Hungary
ON CHERN-LAGRANGE COMPLEX CONNECTION
GHEORGHE MUNTEANU
Abstract. In this note we make a report about the utility of a new nonlinear connection in complex Lagrange space, which generalize the well-known
Chern-Finsler complex connection.
A significant result establish its relation with a complex nonlinear connection previously obtained by us from the variational problem.
1. Introduction.
Giving up the homogeneity condition of the fundamental function in a complex
Finsler space, in last years we have made an approach in a new type of geometry,
called by us the complex Lagrange geometry([9],[10],[11]).
In brief we shall recall here the basic concepts of this geometry.
Let M be a complex manifold, dimC M = n, (U, (z i )) complex coordinates in a
local chart. The complexified bundle of the real tangent bundle T M is decomposed
in any z ∈ U according to (1, 0)− vectors and respectively to (0, 1)− vectors,
TC M = T 0 M ⊕ T 00 M.
The bundle πT : T 0 M → M is holomorphic and T 00 M is its conjugate. The
geometric support of complex Lagrange geometry is the complex manifold T 0 M,
dimC T 0 M = 2n, and the induced complex coordinates in a local chat are denoted
by u = (z i , η i ).
In its turn, the complexified TC (T 0 M ) is decomposed in TC (T 0 M ) = T 0 (T 0 M ) ⊕
00
T (T 0 M ), where T 00 (T 0 M ) = T 0 (T 0 M ) and, therefore, our attention is focused on
the (1, 0)−type vectors, by conjugation are obtained the vectors from Tu00 (T 0 M ).
0
0
0
0
n Let V (To M ) = {ξ ∈ T (T M ) / πT ∗ (ξ) = 0} be the vertical bundle of T M and
∂
˙
a local base in vertical distribution Vu (T 0 M ). An horizontal bundle
∂η i = ∂i
i=1,n
is a supplementary subbundle of V (T 0 M ), i.e. T 0 (T 0 M ) = H(T 0 M ) ⊕ V (T 0 M )
. This determines a distribution N : u → Hu (T 0 M ), called complex nonlinear
connection, in brief (c.n.c.). A local base in the horizontal distribution Hu (T 0 M )
is denoted by:
(1.1)
δ
∂
∂
= i − Nij j
δz i
∂z
∂η
1991 Mathematics Subject Classification. 53B40, 53C60, 53C15.
Key words and phrases. holomorphic bundle, Lagrange space.
237
238
GHEORGHE MUNTEANU
Usually, the functions Nij are called the coefficients of (c.n.c.) and if they are
transformed under the rule:
∂z 0k
∂z 0i k
∂ 2 z 0i k
(1.2)
Nk0i j =
N
−
η
j
∂z
∂z k
∂z j ∂z k
δ
then the base δzi = δi i=1,n , called the adapted base of Nij (c.n.c.), satisfies the
following rule of transformation:
δ
∂z 0j δ
=
i
δz
∂z i δz 0j
A (c.n.c.) determines the following decomposition of TC (T 0 M ) :
(1.3)
(1.4)
TC (T 0 M ) = H(T 0 M ) ⊕ V (T 0 M ) ⊕ H(T 0 M ) ⊕ V (T 0 M )
0
and the corresponding local
n adapted base
o in any u ∈ T M is obtain by conjugation,
briefly being denoted by δi , ∂˙i , δī , ∂˙ī .
Let D be a derivative law on TC (T 0 M ) which preserves the four distributions
from (1.4). In [9] a particular class of this kind of derivative law is considered,
called N − (c.l.c.),(N − complex linear connection), characterized by the fact that
in addition the complex and tangent structures associated to the (c.n.c.) are preserved.
Locally, a N −
(c.l.c.) is determined only by the following set of coefficients
ī
ī
i
i
Ljk , Cjk , Lj̄k , Cj̄k and their conjugates, where:
(1.5)
Dδk δj = Lijk δi ;
i ˙
∂i
D∂˙k ∂˙j = Cjk
Dδk δj̄ = Līj̄k δī ;
ī ˙
D∂˙k ∂˙j̄ = Cj̄k
∂ī
Definition 1.1. A complex Lagrangian on T 0 M is a differentiable function L :
T 0 M → R under the condition gij̄ = ∂ 2 L/∂η i ∂ η̄ j is a nondegenerate metric. The
pair (M, L) is called a complex Lagrange space.
If the function L : (z, η) → L(z, η) is absolutely homogeneous of two degree in
2
respect to η, i.e. L(z, λη) = |λ| L(z, η), the pair (M, L) is said to be a complex
Finsler space. In a complex Finsler space , the Finsler metric gij̄ satisfies in addition
the following formulas which are consequences of Euler theorem concerning the
homogeneity:
∂gkj̄ i
∂gij̄ i
∂L i
∂L
(1.6)
η = L ; gij̄ η i =
;
η =
η = 0 ; gij̄ η i η̄ j = L
∂η i
∂ η̄ j
∂η i
∂η k
The geometry of a complex Lagrange space, particularly complex Finsler one,
depends on the choice of the (c.n.c.)N. Certainly, such geometry must necessary
contain only geometrical objects related to the space metric. Consequently, the
(c.n.c.) is required to be expressed only on complex Lagrange function L.
Initially this problem was solved by us using the variational problem for a complex geodesic([9])
ON CHERN-LAGRANGE COMPLEX CONNECTION
239
Theorem 1.1. Let H j be given by :
Hj =
(1.7)
1 h̄j ∂ 2 L k
g
η
2
∂z k ∂ η̄ h
c
j
Then Nij = ∂Hi are the coefficients of a complex nonlinear connection, determined
∂η
only by the complex Lagrangian L.
In particular , when L = gij̄ η i η̄ j is a complex Finsler metric, and denoting by
∂g
i
kl̄
γjk
= 12 g l̄i ( ∂zjkl̄ + ∂g
∂z j ) the first of the Christoffel symbols corresponding to the
Hermitian metric gij̄ ([3], [6]), from the above theorem it results:
Theorem 1.2. The functions:
c
Nij =
(1.8)
j
1 ∂γ00
i
i
, where γ00
= γjk
ηj ηk
2 ∂η i
are the coefficients of one (c.n.c.) on T 0 M , called the Cartan complex nonlinear
connection of (M, L) Finsler space.
For a given (c.n.c.)Nij , a metric N −(c.l.c.) in respect to the hermitian structure
on T 0 M
G = gij dz i ⊗ dz j + gij δη i ⊗ δη j
(1.9)
is given by ([9]):
c
Lijk =
δg
1 li δgjl
)
g ( k + kl
δz j
δz
2
c
(1.10)
Līj̄k =
c
i
; Cjk
=
∂gj l̄
∂g
1 li ∂gjl
) = g l̄i k
g ( k + kl
j
∂η
∂η
∂η
2
δgkj̄
1 īl δglj̄
g ( k −
)
2
δz
δz l
c
;
ī
=
Cj̄k
∂gkj̄
1 il ∂glj̄
)=0
g ( k −
2
∂η
∂η l
and is called canonical N − (c.l.c.)
In the particular case of complex Finsler spaces, as a rule, is studied a special
connection called the Chern-Finsler complex connection ([1],[2]..):
(1.11)
Nij = g m̄j
∂glm̄ l
δgj m̄
∂gj m̄
i
η ; Lijk = g m̄i k ; Cjk
= g m̄i
i
∂z
δz
∂η k
ī
and Līj̄k = Cj̄k
= 0, which being of (1, 0)−type, assume some facilities in calculus.
On the other hand, the canonical N − (c.l.c.), although is not of (1, 0)−type,
has the advantage of h − (h, h) and v − (v, v) vanishing torsions.
240
GHEORGHE MUNTEANU
2. The Chern-Lagrange complex connection.
Having in mind the dualism of Lagrangian-Hamiltonian principles from classical
mechanic, recently we have made an intensive approach of the geometry of complex
Hamilton spaces ([12]). Firstly, by direct hand calculus and then by geometrical
reasons, we have succeeded in finding one (c.n.c.) in a complex Hamilton space with
a very simple expression and also its utility was proved. Then we have asked to
find its back image by the complex Legendre transformation in complex Lagrange
space . The purpose of the note is not to describe this technique which is quite
ample. We shall make here just an analyze of the obtained result.
KL
Theorem 2.1. The following functions Nij are the coefficients of a (c.n.c.) in the
complex Lagrange space (M, L) :
KL
Nij = g k̄j
(2.1)
∂2L
∂z i ∂ η̄ k
called the complex Chern-Lagrange nonlinear connection.
KL
By direct calculus is proved that the coefficients Nij verify the (1.2) law of
transformation.
Proposition 2.1. The brackets of the adapted base of Chern-Lagrange (c.n.c.) are:
KL
KL
[δj , δk ] = 0 ; [δj , δk̄ ] = δk̄ (Nji )∂˙i − δj (Nji )∂˙ī
h
i
h
i
KL
KL
δj , ∂˙k
= ∂˙k (Nji ) ∂˙i ; δj , ∂˙k̄ = ∂˙k̄ (Nji )∂˙i
h
i
h
i
∂˙j , ∂˙k
= 0 ;
∂˙j , ∂˙k̄ = 0
and [X, Y ] = X̄, Ȳ .
(2.2)
1
Proposition 2.2. If
1
Nij
2
0
is a given (c.n.c.) on T M then
1
Nij = 12
1
∂ N0j
, where
∂η i
1
N0j =Nkj η k , is a (c.n.c.) too, called the spray of Nij .
The proof consist in verifying of (1.2) law of transformation.
1
As a result, from a given (c.n.c.) Nij we can obtain a sequence of (c.n.c.). A
question is when this sequence becomes constant. Usual calculus prove that it is
happening if and only if:
1
1
(2.3)
Nij =
∂ Nkj k
η
∂η i
ON CHERN-LAGRANGE COMPLEX CONNECTION
241
c
Theorem 2.2. In a complex Lagrange space the Nij (c.n.c.) given by (1.7) is the
KL
spray of Nij (c.n.c.).
Also, taking into account the first (2.2) formula of brackets it is obvious our
KL
interest for the Nij (c.n.c.).
c
KL
Proposition 2.3. In a complex Finsler space the Nij (c.n.c.) and Nij (c.n.c.)
coincides..
Therefore, the Chern-Lagrange (c.n.c.) is a natural generalization of ChernFinsler (c.n.c.) on a Lagrange complex space.
In respect to adapted bases of Chern-Lagrange (c.n.c.) we can consider the
KL
Nij −(c.l.c.) which has h − (h, h) and v − (v, v) zero torsions.
Remark. Similar reasons gives the notion of spray for a nonlinear connection
in a real Lagrange space, where a nonlinear connection depending only on the
fundamental function of the space is well-known, [8],pag.160:
Nji =
1
∂2L
∂L
∂Gi
with Gi = g ih ( h k y k − h )
j
∂y
4
∂y ∂x
∂x
1
1
The existence of one nonlinear connection Nji such that Nji be the spray of Nji
is still now an open problem.
References
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242
GHEORGHE MUNTEANU
[13] H.I.Royden, Complex Finsler Spaces, Contemporany Math. Am. Math. Soc., 49 (1986),
p.119-124.
[14] H.Rund,The curvature theory of direction -dependent connection on comlex manifolds, Tensor N.S. 24,1972,p.182-188.
”Transilvania” Univ., Faculty of Science, 2200, Brasov, Romania
E-mail address: gh.munteanu@info.unitbv.ro